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We used wind energy site survey data from Appendix A in Bay et al. (2016) to obtain example survey values. This data recorded plot area in hectares, and observation time in minutes. We converted these to survey effort consistent with the units used by FWS (hr km3) by rescaling plot area (ha) by 0.2 km/ha, and multiplying by the reported observation time in hours.
We also estimated the expansion factor (not reported in Bay et al. (2016)) for each site to produce fatality estimates in numbers of eagles per year. We quantified the relationship between ‘Flight Risk Area’, and predicted ‘Collisions per Annuum’ reported in Bay et al. (2016) using a generalized linear model. First, we divided Collisions per Annuum reported at each site by the site exposure, and mean of the prior collision rate distribution to obtain the site-specific collision rates per eagle activity rate. We modeled this rate as a quadratic funciton of site risk area.
glm(data = Bay_16, (COLLISIONS/(FLIGHT_MIN/EFFORT))/0.002895415 ~ RISK_HA + RISK_HA^2)
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The coefficients from this model were then used as the expansion factor for each site. Thus we calculate the flight risk area as -2.255810^{4} + 2306 * Risk Area + -2.61 * (Risk Area) 2.
Simulation Data
We generated hypothetical values for survey effort based on the minimum requirements provided by U.S. FWS for pre-construction monitoring. FWS requires at least one cylindrical survey plot with radius > 800m and height > 200m, and that plots be surveyed for at least 12 hours per year, for two years. Thus, the minimum values for survey area and time were 0.402 km3, and 24 hrs, respectively. We simulated up to five plots (area = 2.01) in increments of one, and up to 240 hrs in increments of 12, and calculated survey effort for all combinations.
Observed eagle flight time (min) provided in Bay et al. (2016) is a function of survey effort and eagle activity at a site. Therefore, to generate a range of potential exposure values (min/hr*km3), we divided flight time by effort at each survey site to obtain an effort invariant measure of eagle exposure rate, and took a random sample of 20 values. The final simulation data included all combinations of survey effort and eagle exposure rate, which we multiplied to obtain flight minutes, producing 1000 values.
flight <- sample(BAY16$MIN/Bay16$EFFORT, 20)
survey_time <- seq(1, 10, 1)*12*2
survey_area <- seq(0.402, 2.01, 0.402)
df <- expand.grid(TIME = time, AREA = area, eagle_rate = flight)
df$beta <- df$TIME*df$AREA
df$alpha <- df$eagle_rate*df$beta
Model Output
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We generated posterior distributions for predicted eagle fatalities using the FWS Bayesian model. We use the updated FWS priors for collision rate, Collision ~ Beta(9.38, 3224.51), and a prior exposure probability distribution defined by the survey data, Exposure ~ Gamma(a, b), where a represents observed eagle flight minutes, and b represents survey effort, following Bay et al. (2016). We defined a and b for the exposure prior as the mean values from survey data. To obtain posterior estimates, we drew 100,000 random samples from each of these distributions and collected their products to form the posterior distribution of eagle fatality rates.
prediction <- function(iters, alpha, beta){
out <- data.frame(collision = rep(NA,iters),
exposure = rep(NA, iters),
fatality = rep(NA, iters)
)
for(n in 1:iters){
c <- rbeta(1, shape1 = 9.38, shape2 = 3224.51)
e <- rgamma(1, shape = alpha, rate = beta)
f <- c*e
out[n,] <- c(c,e,f)
}
return(out)
}
We generated predictions at each survey site using only data from that site, in which case a and b were set to the observed values, and by updating the prior exposure distribution with observed values following Bay et al. (2016). We applied this same procedure to all 1000 combinations of simulated survey effort and eagle exposure, multiplying the predicted fatality rates by the mean expansion factor derived from survey data to obtain distributions of predicted annual fatalities.
#Predictions integrating Exposure prior and survey data
prediction(100000, a + mean(Bay16$FLIGHT_MIN), b + mean(Bay16$EFFORT))
#Predictions from survey data
prediction(100000, a, b)
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We generated eagle fatality predictions using only simulated or empirical values for observed eagle minutes and survey effort, and by updating the prior probability of exposure using these values. We obtained the posterior exposure distribution as in New et al. (2013) Exposure ~ Gamma(a + minutes, b + effort), where a and b are the shape and rate parameters from the prior distribution.
# simFatal funciton from New et al. (2013) modified to perform multiple iterations.
# BMin: observed number of bird minutes
# Fatal: annual avian fatalities on an operational wind facility
# SmpHrKm: total time and area surveyed for bird minutes
# ExpFac: expansion factor
# aPriExp: alpha parameter for the prior on lambda
# bPriExp: beta parameter for the prior on lambda
# aPriCPr: alpha parameter for the prior on C
# bPriCPr: beta parameter for the prior on C
# iters: number of sampling iterations
simFatal <- function(BMin=-1, Fatal=-1, SmpHrKm, ExpFac, aPriExp=1,
bPriExp=1,aPriCPr=1, bPriCPr=1, iters){
out <- data.frame(collision = rep(NA,iters),
expose = rep(NA, iters),
fatality = rep(NA, iters)
)
# The default of a negative value for BMin or Fatal indicates no data
if(BMin>=0){
aPostExp <- aPriExp + BMin
bPostExp <- bPriExp + SmpHrKm
}else{
aPostExp <- aPriExp
bPostExp <- bPriExp}
# Update the collisions prior
if(Fatal>=0){
aPostCPr <- aPriCPr + Fatal
bPostCPr <- ((rvmean(Exp) * ExpFac) - Fatal) + bPriCPr
}else{
aPostCPr <- aPriCPr
bPostCPr <- bPriCPr}
for(i in 1:iters){
Exp <- rgamma(n=1, aPostExp, bPostExp)
CPr <- rbeta(n=1, aPostCPr, bPostCPr)
Fatalities <- ExpFac * Exp * CPr
out[i,] <- c(CPr, Exp, Fatalities)
}
return(out)
}
estimates <- function(iters, a, b){
out <- simFatal(BMin = a,
Fatal = -1,
SmpHrKm = b,
ExpFac = mean(Bay16$SCALE),
aPriExp = 11.81641,
bPriExp = 9.7656250,
aPriCPr = 1.638029,
bPriCPr = 290.0193,
iters = iters)
fatality <- mean(out$fatality)
q80 <- quantile(out$fatality, c(0.8))
out2 <- simFatal(BMin = a,
Fatal = -1,
SmpHrKm = b,
ExpFac = mean(Bay16$SCALE),
aPriExp = 0,
bPriExp = 0,
aPriCPr = 1.638029,
bPriCPr = 290.0193,
iters = iters)
fatality2 <- mean(out2$fatality)
q82 <- quantile(out2$fatality, 0.8)
return (c("MN_F" = fatality, "U_F" = q80, "MN" = fatality2, "U" = q82))
}
We used an iterative sampling procedure to produce a posterior distribution of annual eagle fatalities. During each iteration, we drew a random sample from the updated exposure distribution and the prior collision probability distribution, and multiplied these values together with a site specific expansion factor. For simulated values, we used the mean expansion factor from the empirical data. For each site and set of simulated values, we performed 100000 iterations of this sampling procedure, and calculated the mean and 80th percentile values of the resulting distribution.
#Predictions integrating Exposure prior and survey data
sim <- plyr::mdply(df[, c(5, 4)], estimates, niters = 100000)
#Predictions from survey data
emp <- vapply(1:nrow(Bay16), function(x){
estimates(100000, Bay16$FLIGHT_MIN[x], BAY16$EFFORT[x])
return (c(fatality, q80, fatality2, q82))
},
USE.NAMES = FALSE, FUN.VALUE = c(0,0,0,0))
)
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